The madeira vine is an environmental weed and there is one growing near The Cottage. It needs to be destroyed.
Recommended is to take down the whole thing and put it in a bin or bucket of water to rot, similar to the approach taken with the khaki burr weed. If the tubers along the vine are not destroyed, each one can sprout into a new vine.
This should be done soon before all the tubers drop and re-sprout.